A seismic survey represents an attempt to image or map the subsurface of the earth by sending energy down into the ground and recording the “echoes” that return from the rock layers below. The source of the down-going sound energy might come, for example, from explosions or seismic vibrators on land, or air guns in marine environments. During a seismic survey, the energy source is placed at various locations near the surface of the earth above a geologic structure of interest. Each time the source is activated, it generates a seismic signal that travels downward through the earth, is reflected, and, upon its return, is recorded at a great many locations on the surface. Multiple source/recording combinations are then combined to create a near continuous profile of the subsurface that can extend for many miles. In a two-dimensional (2-D) seismic survey, the recording locations are generally laid out along a single line, whereas in a three dimensional (3-D) survey the recording locations are distributed across the surface in a grid pattern. In simplest terms, a 2-D seismic line can be thought of as giving a cross sectional picture (vertical slice) of the earth layers as they exist directly beneath the recording locations. A 3-D survey produces a data “cube” or volume that is, at least conceptually, a 3-D picture of the subsurface that lies beneath the survey area. In reality, though, both 2-D and 3-D surveys interrogate some volume of earth lying beneath the area covered by the survey.
A conventional seismic survey is composed of a very large number of individual seismic recordings or traces. In a typical 2-D survey, there will usually be several tens of thousands of traces, whereas in a 3-D survey the number of individual traces may run into the multiple millions of traces. Chapter 1, pages 9-89, of Seismic Data Processing by Ozdogan Yilmaz, Society of Exploration Geophysicists, 1987, contains general information relating to conventional 2-D processing and that disclosure is incorporated herein by reference. General background information pertaining to 3-D data acquisition and processing may be found in Chapter 6, pages 384-427, of Yilmaz, the disclosure of which is also incorporated herein by reference.
A seismic trace is a digital recording of the acoustic energy reflecting from inhomogeneities or discontinuities in the subsurface, a partial reflection occurring each time there is a change in the elastic properties of the subsurface materials. The digital samples are usually acquired at 0.002 second (2 millisecond or “ms”) intervals, although 4 millisecond and 1 millisecond sampling intervals are also common. Each discrete sample in a conventional digital seismic trace is associated with a travel time, and in the case of reflected energy, a two-way travel time from the source to the reflector and back to the surface again, assuming, of course, that the source and receiver are both located on the surface. The seismic energy may also take more varied paths from source to receiver, for example reflecting or scattering multiple times off inhomogeneities, reflecting from the surface of the earth or the bottom of the ocean one or more times, or bending through gradual velocity gradients without reflecting.
Many variations of the conventional source-receiver arrangement are used in practice, e.g. VSP (vertical seismic profile) surveys, ocean bottom surveys, etc. Further, the surface location of every trace in a seismic survey is carefully tracked and is generally made a part of the trace itself (as part of the trace-header information). This allows the seismic information contained within the traces to be later correlated with specific surface and subsurface locations, thereby providing a means for posting and contouring seismic data—and attributes extracted therefrom—on a map (i.e., “mapping”).
Many algorithms exist for transforming the recorded seismic information into a geologically interpretable image. Since seismic data is typically observed (recorded) only at the surface of the earth, whereas the desired image is ideally a volume encompassing all of the interior of the earth that was illuminated by the seismic energy, central to all of these methods is a wavefield-extrapolation engine that computationally simulates the seismic waves propagating inside the earth from source to receiver. As is well known to those of ordinary skill in the art, the transmission, reflection, diffraction, etc., of seismic waves within the earth can be modeled with considerable accuracy by the wave equation, and accordingly wave-equation-based wavefield-extrapolation engines are the method of choice for difficult imaging problems. The wave equation is a partial differential equation that can readily be couched in terms of one, two, or three dimensions. For complex imaging challenges, the constant-density acoustic wave equation extrapolating in time is typically used as the extrapolation engine. Coupled with an imaging condition it yields an image of reflectors inside the earth. Imaging in this way is called “reverse-time migration”. The same extrapolation engine can also be used within an iterative optimization process that attempts to find an earth model that explains all of the seismic information recorded at the receivers. This is called “full-waveform inversion”. Ideally, inversion produces a 3-dimensional volume giving an estimated subsurface wave velocity at each illuminated point within the earth. If the acoustic wave equation is used, which incorporates both velocity and density as medium parameters, inversion may produce a 3-dimensional volume giving both the velocity and density at each point.
If the velocity is not only a function of location inside the earth, but also a function of the direction the waves propagate through a location, an anisotropic wave equation is required to perform the extrapolations for either migration or inversion. Currently, propagating waves anisotropically requires using the much more expensive (often prohibitively expensive) elastic wave equation in the extrapolation engine.
Of course, numerical solutions of the wave equation pose considerable theoretical and practical problems, especially when the computation is performed in three dimensions (“3D”). One particularly vexing problem is that conventional methods of solving the wave equation—except in very simple/unrealistic subsurface configurations—are not exact so that errors and distortions are generally introduced into the calculated results. In practice higher accuracy could only be achieved by using a finer computational mesh, which is often prohibitively expensive.
As is well known in the geophysical prospecting and interpretation arts, there has been a need for a method of extrapolating waves in time using the wave equation that remains accurate without requiring a fine computational mesh, and which can handle anisotropy without requiring an elastic wave equation. Accordingly, it should now be recognized, as was recognized by the present inventors, that there exists, and has existed for some time, a very real need for a method that would address and solve the above-described problems.
Before proceeding to a description of the present invention, however, it should be noted and remembered that the description of the invention which follows, together with the accompanying drawings, should not be construed as limiting the invention to the examples (or preferred embodiments) shown and described. This is so because those skilled in the art to which the invention pertains will be able to devise other forms of this invention within the ambit of the appended claims.